09 - Marginal Analysis

APPLICATIONS OF THE DERIVATIVE TO BUSINESS AND

ECONOMICS

MARGINAL ANALYSIS

Marginal Analysis •  In business and economics, the rates at which

certain quantities are changing often provide useful insight into various economic systems.

•  A manufacturer, for example, is interested not only in the total cost at certain production levels, but is also interested in the rate of change of costs at various productions levels.

•  In economics, marginal analysis is the process of estimating the change in the value of a business-related function like cost, profit, revenue, or consumption, when the independent variable increases by one unit.

Marginal Analysis

•  If f(x) represents a business quantity, its derivative f ’(x) is called the corresponding marginal function.

•  If x is the number of units of a product produced in some time interval, then

•  Total Cost :TC(x)

•  Marginal cost function: TC ‘ (x) = MC(x) •  It gives an estimate of the cost of producing an

additional unit of the product.

•  Marginal cost is the instantaneous rate of change of cost relative to production at given production level.

Marginal Analysis



•  Total Revenue: TR(x)

•  Marginal revenue function: TR ‘ (x) = MR(x) •  It gives an estimate for the additional revenue

generated when an additional unit is sold.

•  Marginal revenue is the instantaneous rate of change of revenue relative to production at given production level.

Marginal Analysis



•  Total Profit: P(x) = TR(x) – TC(x)

•  Marginal profit function: P‘(x) = TR’(x) – TC’(x) •  It gives an estimate for the additional profit

generated when an additional unit is sold.

•  Marginal profit is the instantaneous rate of change of profit relative to production at given production level.

Marginal Cost and Exact Cost

•  Total cost of producing x items: TC(x)

•  Total cost of producing x + 1 items: TC(x + 1)

•  Exact cost of producing the (x + 1)st item: Ø  TC(x + 1) - TC(x)

•  Marginal cost function approximates the exact cost of producing the (x + 1)st item:

•  TC’(x) ≈ TC (x +1) – TC(x) •  Similar interpretations can be made for total

revenue functions and total profit functions

Example 1: •  A company manufactures fuel tanks for

automobiles. The total weekly cost (in hundreds) of producing x tanks is given by

TC(x) = 10 000 + 90x – 0.05x2.

•  a. Find the marginal cost function.

•  b. Find TC’(500), and discuss the various interpretations of this result.

•  c. Find the exact cost of producing the 501st tank, and discuss the relationship between this result and the marginal cost in (b.).

Example 1: Solution •  Total weekly cost (in hundreds) :

TC(x) = 10 000 + 90x – 0.05x2.

•  a. Find the marginal cost function.

•  TC’(x) = 90 – 0.1x •  b. Find TC’(500), and discuss the various

interpretations of this result.

•  TC’(500) = 90 – 0.1(500) = 40 •  At a production level of 500 tanks per week,

total production costs are increasing at the rate of Php 4000 per tank.

•  Php 4000 is the approximate cost of producing the 501st tank.

Example 1: Solution •  Total weekly cost (in hundreds) :

TC(x) = 10 000 + 90x – 0.05x2.

•  TC’(x) = 90 – 0.1x

•  TC’(x) = 90 – 0.1(500) = 40 ⇒ Php 4 000

•  c. Find the exact cost of producing the 501st tank, and discuss the relationship between this result and the marginal cost in (b.).

•  TC(501) – TC(500) = 42 539.95 – 42 500

•  = 39.95

•  Exact cost of producing the 501st tank is

Php 3 995.

Example 2: •  Suppose that the total cost (in thousands)

for manufacturing x items per year is given by TC(x) = 575+ 25x – 0.25x2, 0 ≤ x ≤ 50.

•  a. Find the marginal cost at a production level of x items per year.

•  TC’(x) = 25 – 0.5x •  b. Find the marginal cost at a production level

of 40 items per year, and interpret the results.

•  TC’(40) = 25 – 0.5(40) = 5 •  At a production level of 40 items per year, the

total cost is increasing at the rate of 5,000 per item.

TC(x) = 575+ 25x – 0.25x2, 0 ≤ x ≤ 50

TC(x) = 575+ 25x – 0.25x2, 0 ≤ x ≤ 50.

TC’(x) = 25 – 0.5x

•  c. Use the marginal cost function to approximate the cost of producing the 31st item.

•  TC’(30) = 25 – 0.5(30) = 10 •  The cost of producing the 31st item is

approximately Php10,000.

TC(x) = 575+ 25x – 0.25x2, 0 ≤ x ≤ 50.

TC’(x) = 25 – 0.5x

•  d. Use the total cost function to find the exact cost of producing the 31st item.

•  = TC(31) – TC(30)

•  = 1109.75 – 1100

•  = 9.75 or Php9,750.

Exact cost of producing the 31st item

=( total cost

of producing 31 items)

–

(total cost of

producing 30 items)

A certain company produces rubber balls having a cost function TC(x) = 7000 + 2x, at a production level of 0 ≤ x ≤ 10,000.

The price-demand equation is p = 10 – 0.001x.

a. Find the marginal cost function and interpret.

b. Estimate the cost of producing the 3001st rubber ball.

c. Find the actual cost of producing the 3001st rubber ball.

d. Find the marginal revenue at x = 2000, x = 5000 and x = 7000. Interpret the results.

e. Find the marginal profit at x = 1000, x = 4000 and x = 6000. Interpret the results.

f. Graph TC(x) and TR(x) on the same coordinate axes. Interpret the results.

Example 3:

Example 3:

•  TC(x) = 7000 + 2x, 0 ≤ x ≤ 10,000.

•  p = 10 – 0.001x a. Find the marginal cost function and interpret.

TC’(x) = ?

TC’(x) = 2

Since this is a constant, it costs an additional Php2 to produce one more rubber ball at any production level.

Example 3:

•  TC(x) = 7000 + 2x, 0 ≤ x ≤ 10,000.

•  p = 10 – 0.001x c. Find the actual cost of producing the 3001st rubber ball.

TC(3001) – TC(3000) = ? TC(3001) – TC(3000) = 13002 – 13000 = 2 The actual cost of producing the 3001st rubber

ball is Php 2.

Example 3:

•  TC(x) = 7000 + 2x, 0 ≤ x ≤ 10,000.

•  p = 10 – 0.001x d. Find the marginal revenue at x = 2000, x = 5000 and x = 7000.Interpret the results.

TR(x) = x(10 – 0.001 x)

TR(x) = 10x – 0.001 x2

TR’(x) = 10 – 0.002x TR’ (2000) = 10 – 0.002(2000) = 6 TR’ (5000) = 10 – 0.002(5000) = 0 TR’ (7000) = 10 – 0.002(7000) = -4

Example 3:

•  TC(x) = 7000 + 2x, 0 ≤ x ≤ 10,000.

•  p = 10 – 0.001x d. Find the marginal revenue at x = 2000, x = 5000 and x = 7000.Interpret the results. TR’ (2000) = 6 TR’ (5000) = 0 TR’ (7000) = -4

At 2000 output level, revenue increases as production increases

At 5000 output level, revenue does not change with a small change in production

At 7000 output level, revenue decreases with an increase in production

Approximate changes in revenue per unit change in production

Example 3:

•  TC(x) = 7000 + 2x, 0 ≤ x ≤ 10,000.

•  p = 10 – 0.001x e. Find the marginal profit at x = 1000, x = 4000 and x = 6000.Interpret the results. P(x) = TR(x) – TC(x) P(x) = (10x – 0.001 x2 ) – (7000 + 2x) P(x) = 8x – 0.001 x2 – 7000 P’(x) = 8 – 0.002x P’(1000) = 8 – 0.002(1000) = 6 P’(4000) = 8 – 0.002(4000) = 0 P’(6000) = 8 – 0.002(6000) = -4

Example 3:

•  TC(x) = 7000 + 2x, 0 ≤ x ≤ 10,000.

•  p = 10 – 0.001x d. Find the marginal profit at x = 1000, x = 4000 and x = 6000.Interpret the results. P’(1000) = 6 P’(4000) = 0 P’(6000) = -4

At 1000 output level, profit will be increased if production is increased

At 4000 output level, profit does not change with a small change in production

At 6000 output level, profit will decrease if production is increased

Approximate changes in profit per unit change in production

Graph:

L O S S

(10 000, 0)

(0, 7 000)

LOSS

BREAK-‐EVEN POINT

BREAK-‐EVEN POINT

PROFIT

(1 000, 9 000)

(7 000 , 21 000)

Average Functions •  – the average cost function; gives

the cost of manufacturing or producing each of the x units of output

•  – average variable costs

•  – average fixed costs

•  – average revenue function; gives the contribution of each unit sold to total revenue

xxTCxAC )()( =

€

AVC(x) = VC (x )x

€

AFC(x) = FC (x )x

pxAR xpx

xxTR === )()(

Average Functions

•  – average cost function

•  – marginal average cost

•  – average revenue function

•  – marginal average revenue

•  – average profit function

•  – marginal average profit

•  NOTE:

•  The marginal average cost function must be computed by first finding the average cost function and then finding its derivative.

xxTCxAC )()( =

xxTRxAR )()( =

( )xxTC

dxdxMAC )()( =

( )xxTR

dxdxMAR )()( =

xxTPxAP )()( =

( )xxTP

dxdxMAP )()( =

Example For the total cost function determine the average cost, the average

variable cost, the average fixed cost, the marginal cost, and the marginal average cost functions.

Average cost: Average variable cost: Average fixed cost: Marginal cost: Marginal average cost:

xxTC 525)( +=

€

AC(x) = 25+5xx = 25

x +5

€

AVC(x) = 5xx = 5

€

AFC(x) = 25x

€

MC(x) = 5

2

25)(')(x

xACxMAC −==

EXAMPLE:

•  A small machine shop manufactures a drill bits used in the petroleum industry. The shop manager estimates that the total daily cost ( in tens of pesos) of producing x bits is

TC(x) = 1000 + 25x – 0.1x2

a.  Find the average cost function.

b.  Find the marginal average cost function.

c.  Find AC(10) and MAC(10) and interpret.

d.  Use the results in (c.) to estimate the average cost per bit at a production level of 11 per day.

EXAMPLE:

•  A small machine shop manufactures a drill bits used in the petroleum industry. The shop manager estimates that the total daily cost of producing x bits is

TC(x) = 1000 + 25x – 0.1x2

a.  Find the average cost function.

b.  Find the marginal average cost function.

xxTCxAC )()( =

( )xxTC

dxdxMAC )()( =

xxAC x 1.025)( 0001 −+=

1.00001)( 2 −−

=x

xMAC

EXAMPLE: •  TC(x) = 1000 + 25x – 0.1x2

c. Find AC(10) and MAC(10) and interpret.

At a production level of 10 bits per day, the average cost of

producing a bit is Php124, and this cost is decreasing at

the rate of Php 10.10 per bit.

If production is increased by 1 bit, then the average cost

per bit will decrease by approximately Php 10.10.

Thus, the average cost per bit at a production level of 11

bits per day is approximately 124 – 10.10 = Php 113.90.

?)10( =AC ?)10( =MAC124)10( PhpAC = )10.10()10( PhpMAC −=

Let •  C = national consumption

•  S = savings

•  Y = national income

•  Assuming that the national income is divided only between savings and consumption,

•  ⇒

•  Marginal propensity to consume (MPC)

•  Marginal propensity to save (MPS) dYdCMPC =

dYdSMPS =

MPSMPC +=1SCY +=

EXAMPLE •  Suppose that the savings function is defined by the

equation

•  Determine the values of MPS and MPC when Y = 30 and interpret them when the national income increases by 1 unit from its current level.

•  Solution:

•  when Y = 30

15003.0 2 +−= YYS

106.0 −= YMPS MPSMPC +=1

1)30(06.0 −=MPS8.0=MPS

MPCMPS =−12.0=MPC

EXAMPLE •  Suppose that the savings function is defined by the

equation

•  Determine the values of MPS and MPC when Y = 30 and interpret them when the national income increases by 1 unit from its current level.

•  Solution:

•  If the national income is increased by 1 unit, savings will increase by 0.8 and the consumption will increase by 0.2, that is, people are more likely to save than to spend.

15003.0 2 +−= YYS

MPSMPC +=18.0=MPS 2.0=MPC

EXERCISES: (page 210) 1.  It is estimated that the cost of cementing

a road x kilometers(km.) long is TC(x) = x2+ 6x +4 (in million pesos).

a. Find the average cost function.

b. Find the total cost of cementing a road 5 km. long.

c. Estimate the additional cost entailed by extending a 4km. Road by 1 km.

EXERCISES: (page 210) 2. The total cost of producing x thousand

units of a certain product is given by the equation TC(x) = x3 - x2 + 20x (in thousand pesos).

a. Find the total cost of producing 4000 units of the product.

b. Use the marginal cost function to estimate the additional cost entailed by increasing the production level from 8000 units to 9000 units.

EXERCISES: (page 211) 4. The total cost for the product described

in problem 3 is given by the equation

a.  Set up the profit function.

b.  Evaluate the marginal profit function when x = 70 and interpret the result.

240001)(

2xxxTC −+=

EXERCISES: (page 211) 6. Suppose the savings function is defined

by

Determine the value of MPS and MPC and interpret the results when Y = 30.

100204.0 2 +−= YYS

The cost function is TC(x) = 9000 + 2x and the price-demand equation is p = 25 – 0.01x, 0 ≤ x ≤ 2,500 for the manufacture of umbrellas.

a. Find the marginal cost, average cost, and marginal average cost functions.

b. Find the revenue, marginal revenue, average revenue, and marginal average revenue functions.

c. Find the profit, marginal profit, average profit, and marginal average profit functions.

d. Find the break-even point(s).

e. Evaluate the marginal profit at x = 1000, x = 1150 and x = 1400. Interpret the results.

f. Graph TC(x) and TR(x) on the same coordinate axes and locate regions of profit and loss.

Exercise: